This sentence seems crucial: It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of poi

This sentence seems crucial: It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices. So you chop up a sphere (which has a volume V1) into a finite number of collections of points (which don't have a volume), then assemble the collections of points into a new sphere (which has a volume V2 != V1).